The association between human microbiomes and health has garnered a great deal of scientific and popular attention. By allowing rapid and inexpensive characterization of microbial community composition, modern sequencing has uncovered enormous microbial diversity. Determining the presence versus absence of microbes is insufficient however; we need to understand how dysfunctional microbiomes form and how to repair them. A critical step toward the goal of promoting the assembly of microbial communities that support health is to predict their temporal dynamics. There remains, however, a critical gap: untangling causation from correlation. Simply stated, we are currently unable to interpret the biological and clinical relevance buried within the extreme complexity of microbial communities. Our long-term goal is to advance microbiome research by a) developing new models that capture causality in microbial interactions; and b) developing tools to interpret the relevance of microbial interactions for human health. Before the development of data analysis pipelines, we need to establish theoretical underpinnings upon which to base the methods. We have three aims focused on developing such theory. (1) Develop molecule-mediated models of microbial interactions. Existing statistical approaches for modeling the temporal dynamics of microbiomes are built on assumptions that are rarely valid for microbial communities and thus can be profoundly misleading. Misspecified models may mislead researchers toward poor prediction of dynamics, or worse, prescription of a misguided treatment that enhances rather than inhibits a microbial species of interest?a major problem if the species of interest is a pathogen. We will assess the predictive power of statistical time-series models given realistic molecule-mediated interactions in synthetic data and develop new statistical methods that account for time-varying interactions. (2) Predict stability of a microbiome. Even when interactions that govern microbial population dynamics are well estimated, these interactions may not be directly relevant to human health. Rather, we may want to predict higher-level properties of a microbiome such as its resilience. Resilience?the ability of a microbiome to maintain and recover function in the face of perturbations such as by antibiotics or opportunistic pathogens?is related to the mathematical concept of stability. We will develop new measures to capture the resilience of the microbiome. (3) Predict other high-level microbiome properties. Often a property of the microbiome in its entirety is of interest, such as the ability to regulate pH or metabolize a toxin. Borrowing from population genetic theory, we will develop novel mathematical models to predict the temporal dynamics of traits associated with the microbiome. Together, these aims will greatly enhance our understanding and interpretation of the temporal dynamics of microbial communities, and lay the foundation for our capacity to influence their trajectories toward desired outcomes. This research will provide a critical step in enhancing our ability to assess risk, design synthetic microbial communities to perform tasks, and manipulate microbiomes to promote health.